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Multiple (Linear) Regression - Recap Used to model the relationship between categorical or continuous independent variables and a continuous dependent variable Assumes that this relationship is linear Tells us what effect a one-unit increase in x will have on y using the coefficient (‘B’) What if we have a categorical dependent?...

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Intro To Logistic Regression I Logistic regression allows us to predict the probability of y having a given value based on information from categorical and continuous independent variables Binary logistic model – when categorical dependent has only two response categories (e.g. male/female) Multinomial logistic model – when categorical dependent has more than two response categories (e.g. Lab/Con/LD/Green…) Allows us to calculate how a change in x affects the odds of y e.g. respondents who played games consoles were more likely to be male than female (odds increase of 4)… or… the odds playing a games console were 4 times higher for males than for females This is not the same as ‘likelihood’!

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Intro To Logistic Regression III y = a + b x ‘y’ represents the dependent variable (what we are trying to predict) e.g. income or sex ‘a’ represents the intercept (where the regression line crosses the vertical ‘y’ axis) aka the constant ‘b’ represents the slope of the line (the association between ‘y’ & ‘x’) e.g. how income or sex changes in relation to education or console ownership ‘x’ represents the independent variable (what we are using to predict ‘y’) e.g. years in education or console ownership P(y) = 1/(1 + e - (a + bx) ) Probability Logarithmic Transformation

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Intro To Logistic Regression IV Probability is the mathematical likelihood of a given event occurring i.e. probability of being male or female based on predictor variables Resulting value of the logistic regression equation (in this form) gives a value between 0 and 1 A value close to 0 means that y is very unlikely to have occurred A value close to 1 means that y is very likely to have occurred In our example, the outcome might be that the respondent is male Just as in multiple linear regression, the independent variables are given coefficients These coefficients are interpreted as odds rather than unit increases

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Intro To Logistic Regression V The logarithmic transformation allows us to express a non- linear relationship in a linear way Thus the logistic regression equation expresses the linear regression equation using a logarithmic term (referred to as logit) This overcomes the problem of linearity and avoids violating this assumption Residuals can now be normally distributed (requires dependent to take more and two values!)

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Intro To Logistic Regression VII To transform this logistic curve into a straight line (so we have linearity): PROB(Male) = 1/(1 + e - (a + b ‘Income’) ) LOGIT(Male) = a + b ‘Income’ This is the equation for a straight line!This is the equation for the curve! But both of these are complicated to interpret (mental gymnastics required!) so we talk about interpreting the effect of the independent variables in terms of ‘odds’ ODDS(Male) = exp(a + b ‘Income’) or… ODDS(Male) = exp(a) exp (b ‘Income’) or… ODDS(Male) = exp(a) exp(b) ’Income’ ODDS(Male) = exp(a + b ‘Income’) or… ODDS(Male) = exp(a) exp (b ‘Income’) or… ODDS(Male) = exp(a) exp(b) ’Income’ Because the constant (‘a’) does not change, ‘exp(b)’ tells us the effect of the independent variable on the odds ratio (‘ODDS(Male)’)

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Intro To Logistic Regression VIII Probability: The chance or likelihood of a specific event of outcome Probability: The chance or likelihood of a specific event of outcome Logit: The natural log of the odds Logit: The natural log of the odds Odds: The ratio of the probability that a particular event will occur to the probability that it will not occur Odds: The ratio of the probability that a particular event will occur to the probability that it will not occur Probability of rain tomorrow: 20/31 or 2/3 Odds of rain tomorrow: (Prob. of rain) / (Prob. no rain) or (2/3) / (1/3) or 0.6 / 0.3 or 2:1 or 2 EXAMPLE: There are 20 rainy days in March (out of 31 possible days) Logit of rain tomorrow: LN(ODDS(rain)) or LN(2) or 0.69

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Intro To Logistic Regression IX Now we know what the technique is, how it can be useful and what it can tell us Running the model in SPSS and interpreting coefficients next week Multinomial logistic regression is very similar Don’t worry if you haven’t followed the equations! Rest of today – model design and assumptions

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Assumptions AssumptionIssueRecommendation Sample Size Sample should be large enough to populate categorical predictors. Limited cases in each category may result in failure to converge Use crosstabs at variable selection stage to identify low populated cells, may result in recoding Outliers Cases that are strongly incorrectly predicted may have been poorly explained by the model and misclassified Identify cases through classification table and residuals – use probability threshold scores Independence of Errors Cases of data should not be related i.e. one respondent per dataset, not repeated measures - overdispersion Easy to avoid if the data collection has been conducted properly Multicollinearity Independent variables are highly inter- correlated (continuous) or strongly related to each other (categorical) Use collinearity diagnostics in linear regression model and test high tolerance values using chi-square or correlation Does not assume normal distribution of predictor variables – very useful!

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Choosing Model Variables I Choosing the variables for your model is not guess work! You need to form hypotheses about which independents might be related to the dependent and why Perform hypothesis tests (chi-square, t-tests etc) to ensure that there is a relationship Understand that p-values of around 0.05 may be accepted – there is no hard and fast rule Cell counts for crosstabs must not drop below 5 as this may result in model computation problems (e.g. if independent perfectly explains dependent) Use this opportunity to check for outliers and to identify categorical variables that may need recoding (collapsing to increase cell counts) – start with frequencies These problems are much easier to deal with before running a model

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Choosing Model Variables II Logistic Regression will exclude any cases where one or more of the independent variable values is missing When choosing variables you must look carefully at the amount of missing data – 50% missing data from one independent variable will exclude 50% of sample from analysis This effect can accumulate to unacceptable levels EXAMPLE: In my PhD thesis I designed a multinomial logistic regression model with 22 original variables which excluded 90.56% of cases due to missing data. After excluding 7 of the worst offenders the percentage of included cases rose to 75.01%. This is a big deal!

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Multicollinearity I Multicollinearity is particularly problematic for logistic regression models It occurs when one or more independent variables are related to each other (i.e. not independent!) It tends to reduce or negate the influential effect of either predictor and can also have cumulating effects on the rest of the model It must be prevented at all costs and is more common than you might think – income, education, social class, age, house ownership, political party affiliation…

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Multicollinearity II To test for multicollinearity you need to use the ‘collinearity diagnostics’ available under ‘Linear’ regression in SPSS Eigenvalues – smaller values mean that the model is likely to be less affected by changes to the measured variables Condition Index – the square root of the ration of the largest Eigenvalue to the Eigenvalue of interest, disproportionately large values are indicative of collinearity Variance Proportions – show % of variance of regression coefficient associated with relevant (small) Eigenvalue, more than two high values on the same dimension may be indicative of collinearity (I use =>0.30) As Eigenvalues shrink towards the bottom of the table collineairty tends to appear around the bottom, but similar Eigenvalues will prevent this Use as a diagnostic test – investigate further with chi-square, t-tests or correlation

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Multicollinearity III Collinearity Diagnostics a ModelDimensionEigenvalue Condition Index Variance Proportions (Constant) ethnicity, 2cat (derived) Highest educational qualification Previously stood as a Parliamentary candidate professional association charitable organisation local party in a local pressure group Trade unions Local pressure group Community Groups Personal Friends Business Associates Employers Party Members Party Agents More people seeking selection than seats? Did you apply for more than one seat in 2006 ? STAND3 PAPER3 LikelyC Reputation local public body a. Dependent Variable: USE THIS VAR

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Coding and Dummy Variables Recoding categorical predictors into binaries Sex is a binary (1=male, 0=female recode) E.g. Live in ‘city’, ‘rural’, ‘suburban’ area all in single variable needs recode into dummy variables: – ‘City’ yes/no (1/0) – ‘Rural’ yes/no (1/0) – ‘Suburban’ yes/no (1/0) This allows us to make statements such as “those who lived in a city were less likely to feel safe” and “those who lived in a rural area were more likely to feel safe” Also important for ordinal variables (e.g. highest qualification) as respondents with a degree will also have A-Levels and GCSEs – this is an assumption in a categorical variable with several responses and needs to be made explicit for logistic regression Generally speaking, all categorical variables should be recoded into dummies – SPSS will do this for you but you need to be aware that it is happening (I’ll show you next week)

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Workshop Task Investigate the LFS dataset Select variables for a binary logistic model Use the workshop slides on the portal to help